The Hecke algebra of a Frobenius P-category

TL;DR

This paper introduces a new Hecke algebra associated with Frobenius P-categories, generalizing classical results like the Alperin Fusion Theorem and relating it to the usual Hecke algebra of finite groups.

Contribution

It defines a novel Hecke algebra for Frobenius P-categories and extends key theorems and concepts from group theory to this broader categorical setting.

Findings

Formulation of the Hecke algebra H_F for Frobenius P-categories.

Generalization of the Alperin Fusion Theorem to Frobenius P-categories.

Relationship established between H_F and the classical Hecke algebra of finite groups.

Abstract

We introduce a new avatar of a Frobenius P-category F in the form of a suitable sub-ring H_F of the double Burnside ring of P - called the Hecke algebra of F - where we are able to formulate the generalization to a Frobenius P-category of the Alperin Fusion Theorem, the "canonical decomposition" of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in the chapter 6 of [4], the "basic P X P-sets" in the chapter 21 of [4], and the generalization by Kari Ragnarsson and Radu Stancu to the virtual P X P-sets in [6]. We also explain the relationship with the usual Hecke algebra a of finite group.